State Standards: GLEs/GSEs National Content Standards:
Grade Span Expectations [Math]:
M(N&O)–10–8 Applies properties of numbers to solve problems, to simplify computations, or to compare and contrast the properties of numbers and number systems. M(F&A)–10–1 Identifies, extends, and generalizes a variety of patterns (linear and nonlinear) represented by models, tables, sequences, or graphs to solve problems. M(PRP)–HS–1 Students will use problem-solving strategies to investigate and understand increasingly complex mathematical content and be able to:
• Expand the repertoire of problem-solving strategies and use those strategies in more sophisticated ways.
• Use technology whenever appropriate to solve real-world problems (e.g., personal finance, wages, banking and credit, home improvement problems, measurement, taxes, business situations, purchasing, and transportation).
• Formulate and redefine problem situations as needed to arrive at appropriate conclusions.
Grade Span Expectations [Physics]:
PS3 (7-8) 8: Students demonstrate and understanding of motion by… (8a)… measuring the distance and time for a given moving object and using those values to calculate speed. (8b)… Solving for any unknown in the expression v=d/t given values for the other two variables. (8c)… Differentiating between speed, velocity and acceleration.
PS3 (9-11) 8: Students demonstrate and understanding of forces and motion by… (8a)… Predicting and/or graphing the path of an object in different reference planes and explain how and why it occurs. (8b)… Using modeling, illustrating, and graphing to explain how distance and velocity change over time for a free falling object.
PS3 (Ext) 8: Students demonstrate an understanding of forces and motion by… (8aa)… Using a quantitative representation of how distance and velocity change over time for a free falling object. (8bb)… Using a quantitative representation of the path of an object that has both horizontal and free fall motion. (8cc)… Modeling, illustrating, graphing, and or quantitatively explaining the path of an object that has both horizontal and free fall motion.
Context of the Lesson Where does this lesson fit in the curriculum and instructional context? Is it the opening of a unit or a series of lessons?
This lesson will take place in the middle of a projectile motion unit. Students should have already been exposed to 1 dimensional kinematics problems and they should be proficient with solving problems involving objects moving in either the x or the y direction. They should also know how to break vectors into components using basic trigonometric ratios.
In the lesson preceding this lesson students will have been introduced to projectile motion and they will have worked with several example problems that involve objects moving in both the x and the y directions at the same time. They will have been exposed to the kinematics equations that are used in the x and y directions and they will understand that the two directions can be dealt with independently.
Finally, students should be able to solve mathematical equations involving 1 variable and they should be familiar with solving systems of equations involving 2 unknowns.
This lesson will allow students to apply their knowledge of kinematics while working with a fun computer application involving firing a cannon at various angles and with various initial velocities.
Opportunities to Learn
Differentiation: Materials, Learners and Environments
Plans to differentiate instruction:
In order to differentiate instruction during this particular lesson I have divided the instruction period into two segments. The first segment will involve a classroom discussion about the range of a projectile and will involve some group work. The second segment will involve the use of a computer program. The two segments of the lesson are designed to be beneficial for all types of learners. Traditional learners will find the first section of the lesson particularly beneficial while students that enjoy working cooperatively will find the final part of the lesson beneficial. The visual and kinesthetic learners will particularly enjoy the second activity because they will be given the opportunity to use a computer program that will put their knowledge and their computational skills to the test. In order to differentiate instruction for the more advanced students I will allow them to experiment with the computer program more. I will provide them with additional tasks that they can complete to further their understanding of the material. This will be discussed further in the coming sections. For the students that may be struggling during this lesson I will have time during the group work to provide them with one on one instruction in order to help clear up any misconceptions that they may be having. I will also provide them with some assistance during the activities and help them follow the problems solving steps. Accommodations and modifications: As discussed in previous lesson plans, my physics classes will be comprised of many different students that all have different learning abilities and learning styles. I will allow the more advanced students to work together so that they can complete their work at their own pace and the students who complete work more slowly can take their time with the assignment. In addition to this the students will only need to complete ½ of the worksheet so the more advanced students can move on to complete the rest of the activity while the other students can take their time with the first portion. I will also provide the students with extra time to finish the assignment if they need it. Environment factors: This lesson involves two separate activities. The first activity will be held in the classroom and students will be seated at their desks. We will have a class discussion and then I will allow them to work with their neighbors around them. The second activity will involve group works and I will have previously set up the lab benches around the room (each with its own computer). It is important that the computer be positioned in a way so that the entire group can see it. It is also important that the computer program is already up and running before the class period begins. This will avoid wasting time attempting to fix computer issues during class. The students will also not have the opportunity to look up unrelated things on the internet. Materials: ü Computers ü PHET projectile program http://phet.colorado.edu/en/simulation/projectile-motion ü PHET worksheet
Objectives
Students will demonstrate their understanding of projectile motion by successfully computing the range for various projectiles launched at different angles and from different initial heights.
Instructional Procedures
LAUNCH A:
When the students walk in to the classroom the agenda for the day will already be posted on the whiteboard at the front of the classroom. This will allow us to refer back to this agenda throughout the lesson in order to stay focused and on task. As the student funnel into the classroom, the will take their seats and begin working on the "daily question". The daily question for the day will be "What was the range of a standard cannonball from the 1950's (in yards)". This will get the students thinking about the range of projectiles which will be the focus of the lesson.
At the beginning of this lesson I will have the students take a short (1 question, 10 point) quiz on projectile motion. This quiz can be seen in the attached document. They will be given 15-20 minutes to complete this quiz and those who don’t finish will be allotted extra time. The next portion of this lesson is designed to introduce the students to the idea of the range of a projectile. The students will have already worked with problems involving projectiles launched at different angles but this lesson will be the first exposure that they will have to actually deriving the “range equation”. I will first explain to the students how this lesson fits into the projectile motion unit: “When studying the motion of a projectile it is oftentimes very useful to know how far the projectile can travel if it were launched with an initial velocity at a given angle. If we could make these measurements and calculations really quickly we could become really good at basketball! (…leave time for laughter). Today we are going to investigate how we can find the range of a projectile.” I will begin the lesson by issuing the class a challenge. I will explain to them that they can form their own groups and work on trying to derive an equation for the range of a projectile launched at an angle. The equation that they derive must only be in terms of the initial velocity of the projectile, the initial angle of the projectile, and g (the acceleration due to gravity). For groups that may be struggling with this task I will issue several hints. These hints can be seen in the attached document. “The answer lies in the three equations that you are all so familiar with” “If you were given the initial velocity and an angle could you find how far the object travels. The situation is the same here except we will be working symbolically?” “What do we know about the velocity of the object in the x direction?” “What can you tell me about the acceleration of the object in the y direction?” “What can you tell me about the shape of the objects path? Is there any symmetry? Can this symmetry help us to accomplish anything?” “What must the y velocity of the projectile be at the top of its flight?” “If we knew the total time of flight could I find the range immediately? If I told you I was traveling in the x direction at 5 mph for 5 hours, how far did I travel? This idea applies here as well?” “Which equations would be helpful for me to use in this situations?” “Make sure that we work symbolically! We are given the initial angle and initial velocity that a projectile is launched at but we still want to keep these variables in our equations because we don’t have any specific numbers to use in this problem” I want all of my students to get in the habit of not plugging in numbers into the equations until the very last step. This activity will help them to develop this habit. This will help them eliminate calculator errors and rounding errors and the final answer will be more accurate when they complete a given problem. It is also good practice for higher-level mathematics and physics because those courses are purely symbolic. Once the students have worked for 10-15 minutes on this task I will have the class reconvene as a whole. I will go around and check the equations that the different groups derived and I will choose one group (that arrived at the correct answer) to go to the board and present what they did and what they found. After they present I will ask the class if the group’s work was believable and if they think that the equation is the correct equation. This will result in a class discussion. If no groups did the exercise correctly then I will perform the derivation for them and I will walk them through the steps. We will then have an artistic student write this equation on a poster board and will post it in the room so that students can refer to it when working on problems in the future. EXPLORE A: After this exercise I will instruct the class that they will be split into groups and they will be working with an awesome computer program for the day that deals with cannons! I will distribute the PHET worksheet that I have attached and I will assign the students their groups of 3 (which I will have decided ahead of time). The groups will meet at the specified lab benches around the room and there will be one computer at each bench. If the room does not have the correct number of computers I will have larger groups and the students will have to take turns using the computers. Before starting the activity I will explain the purpose of the activity to the students: “The University of Colorado developed a unique application that deals with projectiles. This computer simulation allows you to control the muzzle velocity and angle of a cannon. I would like for you to play around with this simulation and apply your knowledge of projectile motion to complete this activity. This will help you to better understand how to analyze the motion of a projectile.” This activity is designed so that the students are given an opportunity to “play around” with the program and figure out how it works without me directly telling them. The worksheet walks them through several steps. These steps include using the computer program, doing computational work with paper and pencil, testing these results using the computer, and drawing conclusions from these results. See the attached document for more details on the types of questions that the students will be asked. This activity is very similar to the “range” situation that the students were introduced to in the beginning of the lesson except the projectile drops a certain distance below the starting point and thus travels even further. Students will have to show that they understand how to compensate for this and adjust their equations/derivations accordingly. For students that may be struggling with this task I will only require that they complete the first 12 questions on the worksheet. Therefore they won’t have to worry about completing the entire worksheet in the allotted time and this extension is not necessary for the class curriculum. If they are having trouble working with the computer program I will help them understand how it works and the tools that can be used. If they are having trouble with their computations I will offer them some guiding questions: “What are the “givens” in this particular situation?” “Which equations will help us solve this particular problem?” “What can we use from the first part of the lesson in this activity?” “Would a diagram help in this situation?” For students that may find this task too easy I have included some additional problems on the worksheet that are more challenging. They will be asked to compute the range for several different angles and explain why the range of projectiles launched at two different angles could potentially be the same. They will also have to explain what would happen if air resistance were not disregarded. If they finish the worksheet I will offer them a few extra challenges. “How long will it take for the projectile in problem 1 to reach a point that is 10 meters above it’s starting point?” “A target is 2000m away from a gunman and is 800m above the ground. The gunman fires at the target and the initial velocity of the bullet is 500 m/s. At what angle does he have to fire the bullet in order to successfully hit the target?” This problem is quite challenging and involves solving simultaneous equations so it will be a good challenge for the advanced students. SUMMARIZE/SHARE A: Once all of the groups have finished “exploring” with the program we will reconvene as a class and summarize what they learned by using the program. I will ask the class the following questions to gauge their understanding of the content knowledge. “What did you learn from using this simulation?” “How did you compensate for the fact that the projectile fell farther below the starting point?” “Where did the range equation come into play in this activity” “Where did the fact that the projectile’s motion is symmetric come into play in this activity? Is it still symmetric even though it falls farther below its starting point?” “What would happen if we launched the cannon straight up? Is this realistically what would happen?” “What real world effects have we ignored in ALL of the problems that we have done thus far? Did the simulation account for this?” Based on their responses to these questions I will gauge how well they understand the content and I will judge whether or not they are prepared for the group investigation of Lesson 6 and 7. I will also ask the class if they have any additional questions or concerns about the activity before we conclude the lesson.
Assessment
In order to assess my students during this investigation I will determine whether each student has successfully completed each of the objectives described above. Launch A: During the first portion of the lesson the students will be taking an independent quiz. They will be allowed to use their calculators for this quiz and they will be required to show all of their work in order to receive full credit for the assessment. I will formally assess them on their ability to use their problem solving skills, identify the known and unknown variables, follow the problem solving steps, and mathematically compute the correct answer. During the next part of the lesson, I will walk around the room and help the groups with the task that was assigned to them. In doing so I will informally assess the students on the understanding of projectile motion. I will also assess them on their ability to manipulate equations and solve simultaneous equations. They will also be assessed on their ability to work cooperatively in groups. Explore/Summarize A: During the explore phase of this lesson the students will complete a worksheet in groups relating to a computer program. They will only be required to complete the first 12 problems of this assignment for full credit and will be offered extra credit if they complete more. I will collect each student’s work and will check over their computations to ensure that they are explicitly using the problem solving steps and techniques that I taught them. This will be a form of formal assessment but I will not be grading on whether they are right or wrong. This activity will help me gauge whether or not they are ready for the investigation that I will perform with them the following lesson. If they are not ready then I will not proceed with the unit.
Reflections This section to be completed only if lesson plan is implemented.
Lesson Implementation: Was not able to implement this lesson yet.
Lesson #5: PHET Projectile Motion Simulation
Lesson Title
PHET Projectile Motion Simulation
National Content Standards:
M(N&O)–10–8 Applies properties of numbers to solve problems, to simplify computations, or to compare and contrast the properties of numbers and number systems.
M(F&A)–10–1 Identifies, extends, and generalizes a variety of patterns (linear and nonlinear) represented by models, tables, sequences, or graphs to solve problems.
M(PRP)–HS–1 Students will use problem-solving strategies to investigate and understand increasingly complex mathematical content and be able to:
• Expand the repertoire of problem-solving strategies and use those strategies in more sophisticated ways.
• Use technology whenever appropriate to solve real-world problems (e.g., personal finance, wages, banking and credit, home improvement problems, measurement, taxes, business situations, purchasing, and transportation).
• Formulate and redefine problem situations as needed to arrive at appropriate conclusions.
Grade Span Expectations [Physics]:
PS3 (7-8) 8: Students demonstrate and understanding of motion by…
(8a)… measuring the distance and time for a given moving object and using those values to calculate speed.
(8b)… Solving for any unknown in the expression v=d/t given values for the other two variables.
(8c)… Differentiating between speed, velocity and acceleration.
PS3 (9-11) 8: Students demonstrate and understanding of forces and motion by…
(8a)… Predicting and/or graphing the path of an object in different reference planes and explain how and why it occurs.
(8b)… Using modeling, illustrating, and graphing to explain how distance and velocity change over time for a free falling object.
PS3 (Ext) 8: Students demonstrate an understanding of forces and motion by…
(8aa)… Using a quantitative representation of how distance and velocity change over time for a free falling object.
(8bb)… Using a quantitative representation of the path of an object that has both horizontal and free fall motion.
(8cc)… Modeling, illustrating, graphing, and or quantitatively explaining the path of an object that has both horizontal and free fall motion.
Where does this lesson fit in the curriculum and instructional context? Is it the opening of a unit or a series of lessons?
In the lesson preceding this lesson students will have been introduced to projectile motion and they will have worked with several example problems that involve objects moving in both the x and the y directions at the same time. They will have been exposed to the kinematics equations that are used in the x and y directions and they will understand that the two directions can be dealt with independently.
Finally, students should be able to solve mathematical equations involving 1 variable and they should be familiar with solving systems of equations involving 2 unknowns.
This lesson will allow students to apply their knowledge of kinematics while working with a fun computer application involving firing a cannon at various angles and with various initial velocities.
Differentiation: Materials, Learners and Environments
In order to differentiate instruction during this particular lesson I have divided the instruction period into two segments. The first segment will involve a classroom discussion about the range of a projectile and will involve some group work. The second segment will involve the use of a computer program.
The two segments of the lesson are designed to be beneficial for all types of learners. Traditional learners will find the first section of the lesson particularly beneficial while students that enjoy working cooperatively will find the final part of the lesson beneficial. The visual and kinesthetic learners will particularly enjoy the second activity because they will be given the opportunity to use a computer program that will put their knowledge and their computational skills to the test.
In order to differentiate instruction for the more advanced students I will allow them to experiment with the computer program more. I will provide them with additional tasks that they can complete to further their understanding of the material. This will be discussed further in the coming sections.
For the students that may be struggling during this lesson I will have time during the group work to provide them with one on one instruction in order to help clear up any misconceptions that they may be having. I will also provide them with some assistance during the activities and help them follow the problems solving steps.
Accommodations and modifications:
As discussed in previous lesson plans, my physics classes will be comprised of many different students that all have different learning abilities and learning styles. I will allow the more advanced students to work together so that they can complete their work at their own pace and the students who complete work more slowly can take their time with the assignment. In addition to this the students will only need to complete ½ of the worksheet so the more advanced students can move on to complete the rest of the activity while the other students can take their time with the first portion.
I will also provide the students with extra time to finish the assignment if they need it.
Environment factors:
This lesson involves two separate activities. The first activity will be held in the classroom and students will be seated at their desks. We will have a class discussion and then I will allow them to work with their neighbors around them. The second activity will involve group works and I will have previously set up the lab benches around the room (each with its own computer). It is important that the computer be positioned in a way so that the entire group can see it. It is also important that the computer program is already up and running before the class period begins. This will avoid wasting time attempting to fix computer issues during class. The students will also not have the opportunity to look up unrelated things on the internet.
Materials:
ü Computers
ü PHET projectile program
http://phet.colorado.edu/en/simulation/projectile-motion
ü PHET worksheet
Instructional Procedures
When the students walk in to the classroom the agenda for the day will already be posted on the whiteboard at the front of the classroom. This will allow us to refer back to this agenda throughout the lesson in order to stay focused and on task. As the student funnel into the classroom, the will take their seats and begin working on the "daily question". The daily question for the day will be "What was the range of a standard cannonball from the 1950's (in yards)". This will get the students thinking about the range of projectiles which will be the focus of the lesson.
At the beginning of this lesson I will have the students take a short (1 question, 10 point) quiz on projectile motion. This quiz can be seen in the attached document. They will be given 15-20 minutes to complete this quiz and those who don’t finish will be allotted extra time.
The next portion of this lesson is designed to introduce the students to the idea of the range of a projectile. The students will have already worked with problems involving projectiles launched at different angles but this lesson will be the first exposure that they will have to actually deriving the “range equation”.
I will first explain to the students how this lesson fits into the projectile motion unit:
“When studying the motion of a projectile it is oftentimes very useful to know how far the projectile can travel if it were launched with an initial velocity at a given angle. If we could make these measurements and calculations really quickly we could become really good at basketball! (…leave time for laughter). Today we are going to investigate how we can find the range of a projectile.”
I will begin the lesson by issuing the class a challenge. I will explain to them that they can form their own groups and work on trying to derive an equation for the range of a projectile launched at an angle. The equation that they derive must only be in terms of the initial velocity of the projectile, the initial angle of the projectile, and g (the acceleration due to gravity).
For groups that may be struggling with this task I will issue several hints. These hints can be seen in the attached document.
“The answer lies in the three equations that you are all so familiar with”
“If you were given the initial velocity and an angle could you find how far the object travels. The situation is the same here except we will be working symbolically?”
“What do we know about the velocity of the object in the x direction?”
“What can you tell me about the acceleration of the object in the y direction?”
“What can you tell me about the shape of the objects path? Is there any symmetry? Can this symmetry help us to accomplish anything?”
“What must the y velocity of the projectile be at the top of its flight?”
“If we knew the total time of flight could I find the range immediately? If I told you I was traveling in the x direction at 5 mph for 5 hours, how far did I travel? This idea applies here as well?”
“Which equations would be helpful for me to use in this situations?”
“Make sure that we work symbolically! We are given the initial angle and initial velocity that a projectile is launched at but we still want to keep these variables in our equations because we don’t have any specific numbers to use in this problem”
I want all of my students to get in the habit of not plugging in numbers into the equations until the very last step. This activity will help them to develop this habit. This will help them eliminate calculator errors and rounding errors and the final answer will be more accurate when they complete a given problem. It is also good practice for higher-level mathematics and physics because those courses are purely symbolic.
Once the students have worked for 10-15 minutes on this task I will have the class reconvene as a whole. I will go around and check the equations that the different groups derived and I will choose one group (that arrived at the correct answer) to go to the board and present what they did and what they found. After they present I will ask the class if the group’s work was believable and if they think that the equation is the correct equation. This will result in a class discussion. If no groups did the exercise correctly then I will perform the derivation for them and I will walk them through the steps. We will then have an artistic student write this equation on a poster board and will post it in the room so that students can refer to it when working on problems in the future.
EXPLORE A:
After this exercise I will instruct the class that they will be split into groups and they will be working with an awesome computer program for the day that deals with cannons! I will distribute the PHET worksheet that I have attached and I will assign the students their groups of 3 (which I will have decided ahead of time). The groups will meet at the specified lab benches around the room and there will be one computer at each bench. If the room does not have the correct number of computers I will have larger groups and the students will have to take turns using the computers.
Before starting the activity I will explain the purpose of the activity to the students:
“The University of Colorado developed a unique application that deals with projectiles. This computer simulation allows you to control the muzzle velocity and angle of a cannon. I would like for you to play around with this simulation and apply your knowledge of projectile motion to complete this activity. This will help you to better understand how to analyze the motion of a projectile.”
This activity is designed so that the students are given an opportunity to “play around” with the program and figure out how it works without me directly telling them. The worksheet walks them through several steps. These steps include using the computer program, doing computational work with paper and pencil, testing these results using the computer, and drawing conclusions from these results. See the attached document for more details on the types of questions that the students will be asked. This activity is very similar to the “range” situation that the students were introduced to in the beginning of the lesson except the projectile drops a certain distance below the starting point and thus travels even further. Students will have to show that they understand how to compensate for this and adjust their equations/derivations accordingly.
For students that may be struggling with this task I will only require that they complete the first 12 questions on the worksheet. Therefore they won’t have to worry about completing the entire worksheet in the allotted time and this extension is not necessary for the class curriculum. If they are having trouble working with the computer program I will help them understand how it works and the tools that can be used. If they are having trouble with their computations I will offer them some guiding questions:
“What are the “givens” in this particular situation?”
“Which equations will help us solve this particular problem?”
“What can we use from the first part of the lesson in this activity?”
“Would a diagram help in this situation?”
For students that may find this task too easy I have included some additional problems on the worksheet that are more challenging. They will be asked to compute the range for several different angles and explain why the range of projectiles launched at two different angles could potentially be the same. They will also have to explain what would happen if air resistance were not disregarded. If they finish the worksheet I will offer them a few extra challenges.
“How long will it take for the projectile in problem 1 to reach a point that is 10 meters above it’s starting point?”
“A target is 2000m away from a gunman and is 800m above the ground. The gunman fires at the target and the initial velocity of the bullet is 500 m/s. At what angle does he have to fire the bullet in order to successfully hit the target?”
This problem is quite challenging and involves solving simultaneous equations so it will be a good challenge for the advanced students.
SUMMARIZE/SHARE A:
Once all of the groups have finished “exploring” with the program we will reconvene as a class and summarize what they learned by using the program. I will ask the class the following questions to gauge their understanding of the content knowledge.
“What did you learn from using this simulation?”
“How did you compensate for the fact that the projectile fell farther below the starting point?”
“Where did the range equation come into play in this activity”
“Where did the fact that the projectile’s motion is symmetric come into play in this activity? Is it still symmetric even though it falls farther below its starting point?”
“What would happen if we launched the cannon straight up? Is this realistically what would happen?”
“What real world effects have we ignored in ALL of the problems that we have done thus far? Did the simulation account for this?”
Based on their responses to these questions I will gauge how well they understand the content and I will judge whether or not they are prepared for the group investigation of Lesson 6 and 7. I will also ask the class if they have any additional questions or concerns about the activity before we conclude the lesson.
Launch A:
During the first portion of the lesson the students will be taking an independent quiz. They will be allowed to use their calculators for this quiz and they will be required to show all of their work in order to receive full credit for the assessment. I will formally assess them on their ability to use their problem solving skills, identify the known and unknown variables, follow the problem solving steps, and mathematically compute the correct answer.
During the next part of the lesson, I will walk around the room and help the groups with the task that was assigned to them. In doing so I will informally assess the students on the understanding of projectile motion. I will also assess them on their ability to manipulate equations and solve simultaneous equations. They will also be assessed on their ability to work cooperatively in groups.
Explore/Summarize A:
During the explore phase of this lesson the students will complete a worksheet in groups relating to a computer program. They will only be required to complete the first 12 problems of this assignment for full credit and will be offered extra credit if they complete more. I will collect each student’s work and will check over their computations to ensure that they are explicitly using the problem solving steps and techniques that I taught them. This will be a form of formal assessment but I will not be grading on whether they are right or wrong. This activity will help me gauge whether or not they are ready for the investigation that I will perform with them the following lesson. If they are not ready then I will not proceed with the unit.
This section to be completed only if lesson plan is implemented.